# C307 — Typed affine model and dependency rank

24 September 2026. Exact rational reduction of the C303–C306 interface; K=529.

## Result

Four affine equations in five typed centers have rank4. The598 relation between the17K and8395 endpoints is dependent. Requiring600 between the17K Noah endpoint and the same-Gear SP counterpart of the fixed cross-bridge endpoint adds an independent equation, giving a unique half-integer solution outside the held annual-center register.

## Keep the five roles distinct

| Symbol | Source role | Conditional equation |
|---|---|---|
| q | Signed cumulative LXX_OFF Flood center | q=−5290+i |
| n | Reflected SP Noah selected by17K | n=q+8993 |
| r | Reflected SP Flood selected by8395 | r=q+8395 |
| b | Reflected MT Flood selected by fixed cross-bridge equality | b=−q−2832 |
| p | Same-Gear, same-boundary reflected SP counterpart of b | p=b+650 |

The cross-bridge equality first gives the signed ordinary MT source x=q+2832. Applying the established Mirror gives b=−x. This distinguishes a physical source translation from the orientation reversal needed to express the reflected endpoint. It does not introduce a new Mirror axis.

Subtracting the equations gives

    n−r = 598
    r−p = 2i−3
    n−p = 595+2i.

Thus the1/3-year interior differences observed for N01/N02 in C304 and the599/601 widths in C305 are consequences of the same equations. They should not be counted as independent discoveries or additional constraints.

## Rank and domain

The598 row adds no rank. Adding n−p=600 raises rank to5 and gives

    (q,n,r,b,p) = (−5287.5,3705.5,3107.5,2455.5,3105.5).

This formal rational solution is not a source-admitted annual center. The existence of a rational solution does not license interpolation of the chronology. The actually supported integer rows remain i2 andi3.

For a source component q+ε, the n/r transports carry +ε while b/p carry −ε. The598 difference preserves phase; the n−p component difference carries2ε. Whole-pair set coincidence, center equality and ancestry-preserving component correspondence are therefore separate claims.

Cumulative Cainan460, regular Cainan130, Terah60, the baseline MT/SP650 frame and sojourn215 remain separately typed context operations. Numerical combinations of these constants do not make their historical roles interchangeable.

## Evidence and next decision

Fifteen mathematical/domain checks and eight authentication checks passed. Exact Fraction-based row reduction and the annual ledger replay byte identically. The rank statement concerns this five-variable interface, not the full project graph. Absolute placement and historical necessity remain conditional. Author review only.

C308 will make one fixed alternative-anchor comparison: retain the8464/12696 bridges, regular reflected AD6 and the same72×528 register, while using the already supplied cumulative reflected ExodusAD1446 alongside ConquestAD1406. This tests whether endpoint density and complete-segment coverage respond differently. No new anchor or primer is introduced.